Theorem opnnei 23168

Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)

Assertion

Ref	Expression
opnnei	⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei

Step	Hyp	Ref	Expression
1		0opn 22952	. . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
2	1	adantr 484	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3		eleq1 2849	. . . . 5 ⊢ (𝑆 = ∅ → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
4	3	adantl 485	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
5	2, 4	mpbird 259	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆 ∈ 𝐽)
6		rzal 4445	. . . 4 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
7	6	adantl 485	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
8	5, 7	2thd 267	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9		opnneip 23167	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
10	9	3expia 1133	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑥 ∈ 𝑆 → 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
11	10	ralrimiv 3152	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
12	11	ex 416	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
13	12	adantr 484	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14		df-ne 2957	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15		r19.2z 4450	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
16	15	ex 416	. . . . . 6 ⊢ (𝑆 ≠ ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
17	14, 16	sylbir 237	. . . . 5 ⊢ (¬ 𝑆 = ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18		eqid 2761	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	neii1 23154	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ⊆ ∪ 𝐽)
20	19	ex 416	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
21	20	rexlimdvw 3167	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
22	17, 21	sylan9r 516	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
23	18	ntrss2 23105	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
24	23	adantr 484	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
26	25	snss 4740	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
27	26	ralbii 3107	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28		dfss3 3923	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
29	28	bilanri 510	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
30	27, 29	sylan2br 604	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
31	24, 30	eqssd 3951	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
32	31	ex 416	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
33	25	snss 4740	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 ↔ {𝑥} ⊆ 𝑆)
34		sstr2 3941	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊆ 𝑆 → (𝑆 ⊆ ∪ 𝐽 → {𝑥} ⊆ ∪ 𝐽))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ∪ 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
36	35	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
37	33, 36	biimtrid 244	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
38	37	imp 410	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → {𝑥} ⊆ ∪ 𝐽)
39	18	neiint 23152	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
40	39	3com23 1138	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41	40	3expa 1130	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42	38, 41	syldan 600	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
43	42	ralbidva 3182	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
44	18	isopn3 23114	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
45	32, 43, 44	3imtr4d 296	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
46	45	ex 416	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽)))
47	46	com23 86	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
48	47	adantr 484	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
49	22, 48	mpdd 43	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
50	13, 49	impbid 214	. 2 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
51	8, 50	pm2.61dan 822	1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085   ⊆ wss 3902  ∅c0 4283  {csn 4579  ∪ cuni 4862  ‘cfv 6516  Topctop 22941  intcnt 23065  neicnei 23145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-top 22942  df-ntr 23068  df-nei 23146

This theorem is referenced by:  neiptopreu  23181  flimcf  24030